Twelve—
Theory of Geolocation by Light Levels
Roger D. Hill
ABSTRACT. A technique for determining the location of elephant seals is described. This technique requires an accurate determination of time of dawn and dusk on a daily basis. The time midway between dawn and dusk, the local apparent noon, determines the seal's longitude, and the day length is used to determine latitude. The longitude determination is equally accurate throughout the year and at all latitudes except those with no dawn and dusk events; the latitude determination is most accurate at the solstices and useless at the equinoxes. Other sources of error are the accuracy of the light-level measurement, atmospheric aberration, and the seal's behavior.
Elephant seals present a particular challenge to the researcher who wants to know where they go. Elephant seals generally surface for inadequate periods of time for reliable tracking by the Argos satellite system (Stewart et al. 1989); however, some researchers have had success with this system by mounting the transmitter on the seal's head. Elephant seals also dive so deep that any instrumentation must be solid or have a substantial pressure housing. These two problems combine to make it difficult to track an elephant seal reliably for more than a few months without the transmitter becoming detached. The price of satellite-transmitters is also generally too high for large-scale studies. However, elephant seals are reasonably faithful to their molting and breeding beaches, so that deployments and recovery of memory-based instruments such as time-depth recorders (TDRs) have been quite successful. In early 1989, it was suggested that I attempt to incorporate a geolocation feature into our TDRs. By recording and storing light levels, times of dawn and dusk could retroactively be determined and used to calculate position using standard solar navigational equations.
Fig. 12.1
Circle A is dawn/dusk interface at
Time A.
The Theory
The principle of determination of location is reasonably straightforward: fig. 12.1 shows the earth, with the sun to the right. The bold circle that encompasses the earth is the line between day and night. Note that this circle does not pass through the north or south pole. Figure 12.2 shows the earth approximately 12 hours later. The sun is still to the right, but the earth has rotated by nearly 180° and Circle A from figure 12.1 has moved with it. Circle B is the line that currently divides day from night. If we consider a point on the earth for which the sun was rising in figure 12.1 and is now experiencing sunset, then the point must be on both Circle B and Circle A. The position is, therefore, an intersection of these two circles (one intersection is for A = dawn and B = dusk, the other is for A = dusk and B = dawn). If we know the times of dawn and dusk and the day of year (which affects the tilt of the earth and thus the position of the dawn-dusk circles), then we can theoretically calculate the location of their intersection and, hence, our position.
The standard equations used for solar navigation (Yallop and Hohenkerk 1985; Nautical Almanac Office 1991) predict the time at which a solar event occurs for a given day of year and location on the earth. The solar event of interest here is when the sun is at an azimuth of 96° (the center of the sun is 6° below the horizon). This is known as civil twilight and is when the sun's first light appears (dawn) or last light disappears (dusk). At these times light level is changing fastest, so that the times of these events can be determined from light-level measurements most accurately. Unfortunately, the standard equations yield the inverse of the required information, so an
Fig. 12.2
Circle B is new dawn/dusk interface at
Time B. Circle A has rotated with the
earth from figure 12.1. Location that had
dawn at Time A and dusk at Time B is
at the intersection of the two circles.
iterative process is used to find the location for which predicted times of dawn and dusk coincide with measured values.
The iterative process starts with an estimate of longitude—longitude (°E) = (time of midnight) × 15, where time of midnight is halfway between dusk time and dawn time and all times are measured in Universal Time (UT) or Greenwich Mean Time (GMT)—and an estimate of latitude of –45° or +45°, depending on the previous position. This first guess of position is used to generate dawn and dusk times, which are compared with the observed dawn and dusk times to produce the next location estimate. This process continues until the observed and predicted dawn/dusk times match to the desired degree of accuracy.
Sources of Error
Equinoxes
Our ability to determine latitude fails near the vernal or autumnal equinoxes, as shown in figure 12.3. The dawn/dusk circle now passes through or very close to the north and south poles. This means that for all places on the earth, the dawn and dusk circles (A and B in fig. 12.2) will now be very close or overlap, and the ability to determine latitude is lost. This does not affect the calculation of longitude. (The location is somewhere on the dawn/dusk circles, which are now also circles of longitude because they pass through the poles.) The usefulness of the solar equations for
Fig. 12.3
At the equinoxes, dawn and dusk circles
overlap, so that latitude determination
is not possible.
calculating latitude is effectively determined by the variation of day length (the time between dawn and dusk) with latitude for each day of the year. If a small difference in day length generates a large difference in latitude, then position is difficult to estimate because any small error in assessing day length will cause a large error in the latitude determination. Conversely, if a large change in day length generates a small difference in latitude, then the position calculated will be accurate.
The relationship between time of year, latitude, and day length is summarized in figure 12.4. The accuracy of a latitude determination for a given day and latitude is determined by the slope of day length with latitude. For example, Day 1 shows a good slope between –45° and +45° latitude, indicating the potential for a good estimate of latitude, and a steeper slope between –60° and –45° and between +45° and +60°, indicating the potential for even better latitude estimates. However, at about Day 265, there is almost no slope except very near the poles, indicating a complete inability to determine latitude from day length variations. Figure 12.5 shows an enlargement of figure 12.4 near the autumnal equinox. Note that between Day 252 and Day 281, the day length at both poles is 24 hours. This is because day length is measured from the time that the sun rises above 96° azimuth to when it sets below 96° azimuth (actually about 7 months at each pole). Had we chosen an azimuth of 90°, day length would have changed from 0 to 24 hours (or vice versa) at both poles on the same day. A side effect of using an azimuth of 96° is that for Days 252 through 281 (and for the equivalent days near the vernal equinox), a measured dawn and dusk time will generate two locations. The correct location must be chosen by comparison to previous locations. Although the quality of latitude determinations near the equinoxes is generally poor, studying the
Fig. 12.4
Day length as a function of day of year and latitude.
Fig. 12.5
Day length as a function of day of year and latitude near the autumnal equinox.
Fig. 12.6
Error in latitude determination caused by a 4-minute error in day length measurement
as a function of day of year and latitude.
gradient of day length with latitude (fig. 12.5) shows that a latitude determination will be possible in some ranges of latitude and day of year (e.g., Day 249, latitude + 45° to + 75°).
An alternate way of displaying this relationship between day of year, latitude, and day length is shown in figure 12.6. A theoretical uncertainty in the accuracy of a location (error) is plotted against latitude and day of year. The error is the range of latitude that is generated by moving dawn and dusk times by ± 4 minutes. An accuracy of ± 2 minutes is the limit of accuracy in observing dawn or dusk due to atmospheric phenomenon, and this has been doubled to reflect other likely errors. The error in latitude has been truncated at 20° for clarity. The slightly skewed nature of the two equinox "ridges" is again caused by using a dawn/dusk azimuth of 96°; had we used 90°, the ridges would not be twisted. The "plateaus" between the ridges show the days and locations where good locations can be expected from a measurement of light levels.
The simplest solution for reducing the ambiguity in latitude near the equinoxes is to use some other measurement to fix the latitude. The obvious choice is surface seawater temperature, which varies considerably with lati-
tude and is available on a week-to-week basis from a combination of satellite imagery and oceanographic buoys. In the waters surrounding the United States, these data are compiled by the National Marine Fisheries Service and the National Weather Service. Latitude is found by determining which locations on the known line of longitude have the measured surface seawater temperature.
Accuracy of the Light-level Measurements
There are several sources of potential error in recording the light-level measurements, only some of which can be controlled. The time at which the light-level measurements were taken must be known accurately; one minute of inaccuracy in the estimate of dawn and dusk times will generate an error of 0.25° of longitude and an error in latitude of about 25% of the error shown in figure 12.5. To minimize the timing errors, users must carefully set the recorder's clock before deployment and note any time error on retrieval. The analysis program must then use this error data and adjust all time measurements accordingly. The magnitude of the light level will change over many orders of magnitude between night and day, so it should be recorded on a logarithmic scale and carefully calibrated so that it will not "peg-out" in bright sunshine or complete dark. Since one will have no control over the orientation of the light sensor when it is collecting data, it should be responsive to light over wide angles and be positioned so that it will generally point up. Obviously, none of these suggestions will help if the study animal is on shore, on its back, and the light sensor is buried in sand. Such data points must be excluded at analysis time.
Atmospheric Aberration
Light does not generally pass through the earth's atmosphere in straight lines; it bends when it encounters thermal or pressure gradients. For this reason, it is generally considered impossible to measure the time of dawn or dusk to an accuracy of greater than 2 minutes, even if one is observing the sun directly rather than measuring ambient light levels (C. Acton pers. comm.). This is a major source of error in this type of navigation; 2 minutes of inaccuracy in the estimate of dawn and dusk times will generate an error of 0.5° of longitude and an error in latitude of about 50% of the error shown in figure 12.5. Some compensation for hot and cold and high and low pressure days can be applied to the navigational equations if one has these data.
Animal Diving at Dawn or Dusk
Elephant seals are known to perform 20-minute dives alternating with 3-minute surface times for many hours at a time (Le Boeuf et al. 1986, 1988,
1989; Stewart and DeLong 1990; DeLong and Stewart 1991). With behavior such as this, a light sensor will probably be at depth during the actual dawn and dusk times. The analysis of light levels must cope with this problem, and, generally, an interpolation technique around the times of dawn or dusk will work well to determine the actual dawn or dusk times. If there is an extended dive at dawn or dusk (greater than 30 minutes), then determination of the dawn or dusk times will not be possible.
Animal Moving between Dawn and Dusk
The above analysis assumes that the animal does not change its location between dawn and dusk. If it does move, the error induced depends on the direction it moves. If the animal moves along Circle A in figures 12.1 and 12.2 between dawn and dusk, then the movement will have no effect on the accuracy of the location, but the location given will be that of the animal at dusk, not some median position. Other directions of movement will have other results. Generally, large errors will only occur when the animal is covering large distances per day, and under these circumstances, a larger locational error is more acceptable. The analysis program could also minimize this error by performing the locational analysis from dusk to dawn (rather than dawn to dusk) when day length is greater than 12 hours.
The Equations
The equations used to predict dawn and dusk times also contain some inherent inaccuracies, as much as ± 2 minutes under certain circumstances. I have been unable to determine for which combinations of day and latitude these inaccuracies are worst, but it seems reasonable that the inaccuracies are going to be most severe where day length changes very rapidly with latitude. These are the same circumstances that give us inherently better locational accuracy. If this is true, then error from the equations will cancel some (or all) of the improved accuracy generated by rapidly changing day lengths. Until better equations can be provided to predict dawn and dusk times, some allowance for errors in the equations must be made.
Presentation of Errors
Since both latitude and day of year will have a profound effect on the accuracy of the latitude determination, it is important that positions generated from observed dawn/dusk times be provided with error estimates. Ideally, all locations should be plotted on a map using rectangles that indicate the limits of the animal's position to a given level of certainty. It should be noted that the center of such rectangles will not necessarily represent the likeliest location of the animal.
Fig. 12.7
Example of analysis program's graphical output. The light-level curve is shown with
markers for derived dawn/dusk times and the calculated position range.
Practical Considerations
Wildlife Computers makes TDRs with temperature- and light-sensing options for use on diving animals. The TDR stores surface-seawater-temperature (SST) and light-level (LL) data when the instrument is at (or very near) the surface. These data are decoded and used by an analysis program to provide locations. The SST and LL data are stored whenever the study animal surfaces but generally no more frequently than every 15 minutes. These light-level data are extracted, plotted, and used to determine the times of dawn and dusk (see fig. 12.7). We have determined that dawn and dusk (sun is at an azimuth of 96°) correspond to a light level equal to the nighttime light level plus 5% of the difference between the night and day levels, when LL data are shown on a logarithmic scale.
It should come as no surprise that some of the errors inherent in attempting to geolocate a swimming or diving animal using light levels are unavoidable. Others, such as temperature effects and interpolation requirements, can be minimized in the analysis software. Our best efforts generally yield locations for elephant seals that are about ± 1 degree in latitude and longitude. Although not as precise as one might like, these location data have greatly expanded our knowledge of the foraging migrations of both southern and northern elephant seals (DeLong, Stewart, and Hill 1992).
References
DeLong, R. L., and B. S. Stewart. 1991. Diving patterns of northern elephant seal bulls. Marine Mammal Science 7: 369–384.
DeLong, R. L., B. S. Stewart, and R. D. Hill. 1992. Documenting migrations of northern elephant seals using day length. Marine Mammal Science 8: 155–159.
Le Boeuf, B. J., D. P. Costa, A. C. Huntley, and S. D. Feldkamp. 1988. Continuous, deep diving in female northern elephant seals, Mirounga angustirostris. Canadian Journal of Zoology 66: 446–458.
Le Boeuf, B. J., D. P. Costa, A. C. Huntley, G. L. Kooyman, and R. W. Davis. 1986. Pattern and depth of dives in northern elephant seals, Mirounga angustirostris. Journal of Zoology, London 208: 1–7.
Le Boeuf, B. J., Y. Naito, A. C. Huntley, and T. Asaga. 1989. Prolonged, continuous, deep diving by northern elephant seals. Canadian Journal of Zoology 67: 2514–2519.
Nautical Almanac Office. 1991. Almanac for Computers . Washington, D.C.: United States Naval Observatory.
Stewart, B. S., and R. L. DeLong. 1990. Sexual differences in migrations and foraging behavior of northern elephant seals. American Zoologist 30: 44A.
Stewart, B. S., S. Leatherwood, P. K. Yochem, and M.-P. Heide-Jorgensen. 1989. Prospects for tracking pinnipeds at sea using the Argos DCLS: Insights from studies of free-ranging harbor and ringed seals. In Proceedings of the 1989 North American Argos Users Conference and Exhibit , Landover, Maryland, 193–203.
Yallop, B. D., and C. Y. Hohenkerk. 1985. Compact Data for Navigation and Astronomy for the Years 1986–1990 . London: Her Majesty's Stationery Office.